Quadratic Extensions, Gaussian Integers

Gaussian Integers

Adjoin i, the square root of -1, to Q and to Z. The ring extension is called the gaussian integers, and a geometric argument proves this ring is a ufd. It is therefore integrally closed in its fraction field Q[i]. We have produced our first quadratic extension, along with its integral ring.

Since Z[i] is an integral ring over a pid, it is dedekind, and since it is dedekind and a ufd, it is a pid. Prime ideals and prime elements coincide.

The norm of a+bi is the product of its conjugates, which is a²+b². Remember that norm and product commute in any ring extension. If u is a unit, write uv = 1 and take norms to show |u| is a unit. Conversely, if |u| is a unit then the product of the conjugates of u, including u itself, is a unit, which makes u a unit.

If |x| is prime in Z then x is prime in Z[i]. Write x = yz and take norms, and |y| or |z| is a unit, whence y or z is a unit. These properties are used to determine when the prime p (in the integers) factors in the gaussian integers. The factorization is completely characterized. In summary, 2 factors into 1+i times itself, p = 1 mod 4 factors into two distinct conjugate primes, and p = 3 mod 4 remains prime in Z[i].

Since prime elements and prime ideals coincide, the splitting problem has been solved for the gaussian integers. There are three cases, for p = 2, 1 mod 4, and 3 mod 4, as described above. You may want to confirm the degree equation in each case. Multiplicity times ramification degree times residue degree = 2.

Since the splitting problem does not change with localization, we know how p factors in Z[i] localized about p. These are the gaussian rationals without p in the denominator.